This paper presents a detailed derivation and description of a new method for solving equality constrained optimization problem. The new method is based upon the quadratic penalty function, but uses orthogonal transformations, derived from the Jacobian matrix of the constraints, to deal with the numerical ill-conditioning that affects the methods of this type.
At each iteration of the new algorithm, the orthogonal search direction is determined by a quasi-Newton method which can avoid the necessity of solving a set of equations and the step-length is chosen by a Armijo line search. The matrix which approaches the inverse of the projected Hessian of composite function is updated by means of the BFGS formula from iteration to iteration. As the penalty parameter approaches zero, the projected inverse Hessian has special structure which can guarantee us to obtain the search direction accurately even if the Hessian of composite function is ill-conditioned in the former penalty function methods.

This paper summarizes the results of comparative testing of (1) Wilf's global bisection method, (2) the Laguerre method, (3) the companion matrix eigenvalue method, (4) the companion matrix eigenvalue method with balancing, and (5) the Jenkens-Traub method, all of which are methods for finding the zeros of polynomials. The test set of polynomials used are those suggested by [5]. The methods were compared on each test polynomials on the basis of the accuracy of the computed roots and the CPU time required to numerically compute all roots.

It is showed that, for many commonly used difference models on hyperbolic initial boundary value problems, the necessary and sufficient condition for GKS-stability is a necessary condition for $l_2$-stability.

The aim of this conjoint paper is to discuss the problem of the best rational approximation with interpolating constraints. In part I, we give the necessary and sufficient conditions for the existence of the best rational approximations and establish some characterization theorem for such approximations. The problems of uniqueness, the properties for the set of best approximations, strong uniqueness and continuity of best approximation operator are considered in Part II. The results obtained in this paper are the completion and extension of those given in [1].

This paper is a continuation of [2]. In this part of the conjoint paper, we establish some results about uniqueness /non-uniqueness, the properties for the set of best approximants, strong uniqueness and continuity of the best approximation operator.  

The symplectic collocation schemes, which are based on the framework established by Feng Kang [1], are proposed for numerical solution of Hamiltonian systems. The sufficient and necessary conditions for various collocation schemes to be symplectic are obtained. Some examples of symplectic collocation schemes are also given.

In [2], general approximation results for the solution in a neighborhood of a simple limit point are given. In this paper we give projective approximation results for the solutions in a neighborhood of a double limit point. Application of these results to a nonlinear partial differential equation and numerical results are given.

Theoretical time step constraints of semi-implicit schemes are known to be more restrictive than should be in practice. We intend to alleviate the constraints with more smoothness assumptions on the solutions. By introducing a new scheme with modification on the treatment of the nonlinear term, we are able to prove that the scheme is unconditionally stable and convergent. Furthermore, we show that the modified scheme and the original semi-implicit one are equivalent under a weak condition on the time step and the number of space discretization points.